The Erdős–Gallai Theorem states that for , any -vertex graph with no cycle of length at least has at most edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If is a 2-connected -vertex graph with no cycle of length at least , then , where . Furthermore, Kopylov presented the two possible extremal graphs, one with edges and one with edges.
In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for odd and all , every -vertex 2-connected graph with no cycle of length at least is a subgraph of one of the two extremal graphs or . The upper bound for here is tight.